Excilus has some good points, but there a number of traps for new players if you take this approach too literally.
Firstly, there are 16 civs and only 5 of them start with the ability to build granaries. So the majority of you wont be able to build them until after youve researched Pottery.
He also suggests that you shouldnt go warrior, settler, settler, settler but doesnt say why. Generally the answer is that (as you need a city with a population of 3) the turns left to build the settler can race down to 1 more and then stop while the population catches up. This wastes potential production time. (The way to assess this is by looking at the 3 figures on the city name box).
Warrior, settler, settler can actually work fine if you have the right starting conditions even settler, settler, settler can sometimes pay off. The key is in understanding all the inter-relationships between shield and food production and the potential of the site you have chosen. A good fast start will benefit pretty much any long term goal, and if you can get those first few settlers out quickly you will soon have a crop of other cities to help produce things. Your first citys build order can be changed considerably as soon as it gets some support.
I dont believe that there is necessarily an overall ideal build order for faster expansion, or anything else (Civ 3 is too complex for that). But there are a variety of better build orders to suit the site you are on, the civ you are playing, and your ultimate overall goals E.g. if you dont have a military campaign in mind you may not want to make as many early military units. There are lots of paths, and none of them seem to work for every occasion.
The computer will also choose which tiles your new citizens start working on, so if it doesnt suit your objectives then change it. You may want to maximise production of food, shields, or even commerce, depending on your goals. Of course you wont always find that a beneficial change is available but, depending on the site, it can sometimes make a big difference.
And dont automatically think that you have failed because your city build rate didnt always match the AI. The AI has been given a fairly snappy build rate (plus some free starting units at harder levels) presumably because the designers feel that the player should be able to balance this by beating the AI regularly in other strategic matters. It might build quickly but it doesnt always build smart, or necessarily know what best to do with the cities once its got them. You may get a lousy start position (e.g. stuck with a lot of jungle) where you are unlikely to match the early build rate of some of the AI civs no matter what you do.
I hope that this sounds constructive, rather than just picky, but I do believe that the best thing that any player can do is to study all the details of the equation rather than rely on a formula approach. Its better than a build and hope style, but it can also lead to a lot of missed opportunities in a game as complex Civ3.
For some more thoughts on this including fairly detailed posts from Anglophile and others you might like to check out the thread below, (where Excilus posted the same suggestions as above).
http://forums.civfanatics.com/showthread.php?s=&threadid=11753